Solution Of Elasticity Equations Research Articles

A Micromechanics Pressurization Model for Cookoff

AbstractA micromechanics pressurization (MMP) model has been derived for explosive decomposition models that are pressure‐dependent. The model includes volumetric thermal strain and internal pressurization using well‐known solutions of elastic equations that include displacement of the condensed phase. The model is based on observations of a heated, high‐density, plastic bonded explosive (PBX) containing 95 wt% triaminotrinitrobenzene (TATB) with 5 wt% chlorotrifluoroethylene/vinylidene fluoride binder (Kel‐F). The model was developed for explosives that are either permeable or impermeable to decomposition gases. The MMP model is based on pore mechanics which describe reaction nucleation, decomposition chemistry, and elastic volumetric expansion. The model accounts for the expansion or swelling of the explosive into the surrounding gas‐filled ullage space. The pressurization model was used in conjunction with a simple decomposition model to determine ignition time and internal temperatures for the TATB‐based explosive at 1881 kg/m3. The MMP model was used to predict pressure, specific surface area, and gas volume fraction. A Latin hypercube sensitivity analysis showed that prediction of ignition time was most sensitive to the maximum pore pressure which defines the threshold between permeable and impermeable explosive layers. The MMP model coupled to a pressure‐dependent chemistry model can predict accurate ignition times for high‐density PBX's exposed to high temperatures and may be useful for more general application scenarios.

Method for finding approximate solutions of elasticity equations using spline wavelets

Method for finding approximate solutions of elasticity equations using spline wavelets

Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.

An exact theory of the bending of transversely inextensible elastic plates

In this article, the exact solution of elasticity equations for transversally inextensible plates obtained using a symbolic integration method is given. In the limiting case, the theory is shown to lead to the Mindlin plate model. The theory is also applied to several cases, including the torsion of a rectangular plate, a rectangular plate under a double sinusoidal load, a strip under linearly variable pressure, a disk under uniform pressure, and an infinite plate with a circular hole subject to cylindrical bending. For the last case, it is shown by asymptotic expansion that the maximum circumferential stress concentration factor tends to a value of 3 for a very small hole.

Willmore surfaces in 3-sphere foliated by circles

Abstract In this paper, we can completely determine the conformal class of Willmore circle-foliated surfaces by a solution of elastic equation and a constant. Moreover, the circle-foliated minimal surfaces in three space forms are also classified.

Structure of polynomial solutions of elasticity equations in stresses

For an isotropic elasticity problem in stresses in three-dimensional space minus the origin, we study solutions that have the singularity 1/r2 and, after the multiplication by r2, polynomially depend on the direction cosines. In this polynomial class, for the equilibrium equation we write out the general solution that is a statically admissible (in the sense of Castigliano) solution of the Kelvin problem. We show that if one or several Beltrami equations are not satisfied, then the classical Kelvin solution becomes nonunique. A method for constructing nonunique solutions of this kind is given. The equivalence of various statements of the elasticity problem in stresses is discussed. For the problem on the action of a lumped force at the vertex of an arbitrary conical elastic body, we write out the exact solution in stresses for the case of an incompressible material. The solution for a compressible material is represented in the form of series in a parameter characterizing the deviation of the Poisson ratio from 1/2. We obtain iterative chains of problems in stresses and conditions for the finiteness of these chains. We also analyze the realizability of a linear-fractional dependence of the solution in stresses on the Poisson ratio.

Calculation of Atom Configuration and Characteristic of Vacancy in bcc Lattice of α-Fe

This work is devoted to the simulation of atom configurations in bcc metals near the point defect using the molecular static method. The values of migration and formation volumes are very sensitive to the atomic structure in the vicinity of a defect, which makes it necessary to consider a large number of atoms in the computation cell and to take into account an elastic matrix around the cell. We have developed the new model taking into consideration these factors. It allows defining the “fine structure” of displacement atoms near the point defect. The atoms of third zone were embedded in an elastic continuum. The displacement of each atom embedded in an elastic continuum was defined as the first and the second terms in solution of elastic equation. In the framework of this model we calculated the formation and migration energies and volumes of defect. Also we take into consideration that the energy of system (in particular the system with defect) depends on the external pressure. This dependence gives an addition to the values of migration and formation volumes.

Generalized finite element methods for three-dimensional structural mechanics problems

The present paper summarizes the generalized finite element method formulation and demonstrates some of its advantages over traditional finite element methods to solve complex, three-dimensional (3D) structural mechanics problems. The structure of the stiffness matrix in the GFEM is compared to the corresponding FEM matrix. The performance of the GFEM and FEM in the solution of a 3D elasticity problem is also compared. The construction of p-orthotropic approximations on tetrahedral meshes and the use of a-priori knowledge about the solution of elasticity equations in three-dimensions are also presented.

Global weak solutions for elastic equations with damping and different end states

In this paper, we study global weak solutions for elastic equations with damping using the compensated compactness method. When the two end states at ± ∞ are not equal, the selfsimilar solutions for the corresponding parabolic equation are used to get the entropic estimates for both the L∞ and L2 cases.